Modeling Games in GDL

The main goal of this class is to learn how to model turn games in a rule based language. After finishing the class, student should be able to model simple games and point out the difficulties related with this task.

1 Preliminaries

GDL (Game Description Language) is a language designed to describe turn games. It was created to be used in so called General Game Playing, a subdomain of the AI, focused on creating bots able to play in any game that can be formally described.

2 Modeling the World

There are two main approaches to model the world of the game:

coarse-grained — where every state of the game is a state in a giant state machine. Every unique state corresponds to one state in the machine. The state machine may grow quite impressive. Beside that state doesn't have any structure so is no really helpful if you want to exploit it to solve the problem in an intelligent manner.

fine-grained — state of the world is defined by so called fluents, which can be treated as variables. The state here is structured (the current values of the fluents) which we may use.

GDL belongs to the fine-grained family, so there are:

fluents (actions made by the players are a special kind of fluent

game rules, which are represented by… rules

3 Example: Tic-Tac-Toe

We will show the basic features of the GDL on the example of the popular tic-tac-toe game.

Players

Let's start with players “nought” and “cross”. There is a special fact role just to do that:

role(nought)
role(cross)

Syntax is the same as in the Prolog (or Datalog) language with the only difference, that there is no dot at the end of the sentece. All constants start with a lower-cased letter and variables are upper-cased.

Fluents

Now we should define fluents that make up the game state. In a Tic-Tac-Toe game there are 29 fleunts:

1 fluent per every cell, telling if there is a cross (x) in the cell (9 fluents)

1 fluent per every cell, telling if there is a nought (o) in the cell (9 fluents)

1 fluent per every cell, telling if the cell is empty (blank - b) (9 fluents)

1 fluent per player, telling if it is his turn to play (2 fluents).

To define the fluents one uses the base facts:

Here :- reprenst implication ←. & is a logical and. In natural language the first line means: “if M and N represent cells' indexes, then there is a fluent for cell with (M, N) coordinates, telling there is a cross in it”. If we didn't use rules, we could write all the fluents by hand:

In other words - player may mark a cell only if it is empty and it is his turn. If it is not his turn, he can only perform the 'noop' action.

Initial State

Now we have to define the initial state of the game (draw the empty board). The init predicate serves this purpose. For the tic-tac-toe every cell should be empty, and the nought player should play first.

init(cell(M,N,b)):- index(M)& index(N)
init(control(nought))

State Changes

Now the most difficult part — how does the state evolve in the game. We assume that the new state is created after the very move and depends only on the previous state and actions made by the players. On could notice the frame problem occurring here. The etymology of the frame is the movie tape, where every frame differs in details from the previous one. Similarly here, we would like the next state to be exactly like the previous one except some changes made explicitly by the players. The problem is that in our case we have also to model lack of changes :(

To define the state evolution we use the next predicate, defining the fluent state in the next turn. The 'does' predicate check what actions were made by the players during the current turn.

if the cell is empty and the 'cross' player marked it then there will be a 'x' in this cell next turn

if the cell is empty and the 'nought' player marked it then there will be an 'o' in this cell next turn

if the cell wasn't empty (distinct checks equality of the arguments), then it will stay the same next turn

if the cell was empty and wasn't mark this turn, it will be still empty next turn (two rules!)

if the current turn belongs to the the 'nought' player, next will belong to the 'cross' player

if the current turn belongs to the the 'cross' player, next will belong to the 'nought' player

It's worth to note, that actions 3 and 4 are designed only to cope with the frame problem.

Game Ending Conditions

Every player knows that there are two ways to finish the game: make a line or fill all the available cells. Now we will check if there was a line made by the player.

The true(line(naught)) query will tell if there is a line made of the 'o's on the board.

Now we can define the ending conditions with the terminal predicate:

Game ends when:

there is a line of crosses

there is a line of naughts

there is no empty cell

The '~' operator means negation. Similarly to the Prolog it follows the closed world assumption, ie. sentence is false if can't prove otherwise. In GDL ther are additional restrictions we want worry about for now (more info here).

Winning Conditions

Bots don't have feelings but we still should give them prize for the good game. And we do this with the goal predicate:

According to the rules above winner gets 100 (so much win!), in case of tie only 50, and loser gets 0. We don't what this number represent but it has to be something good like cookies.

There is no arithmetics in GLD — numbers are only used to define indexes, prizes, time, etc.

End

That's all folks, you've just modeled a simple tic-tac-toe game.

3.1 Assignment 1

Let's take an even simpler game:

Answer the following questions:

how many player is there?

how many fluents are defined?

how many actions are there?

how many legal moves does the 'white' player have in the initial state of the game?

how many fluents are true in the initial state of the game?

assume in the first turn, the white player performed the 'a' action and the 'black' player action 'd'. How many fluents are true in the second turn?

what is the smaller number of turns required to end the game?

does the game always end? (is it possible to loop the game indifinetely?)

3.2 Assignment 2

Please model the 'crosses and crosses' game. The rules are similar to the tic-tac-to, again we have nine cells, but both players use the crosses. Players loses when he first creates the line.

3.3 Assignment 3

Please model the “Rock Paper Scissors Lizard Spock” game. You can learn the rules from this video. The image on the right is also a good reference. You can practice it with a colleague or like a modern man: online....

4 Knowledge Interchange Format

Everything we learned so far is true, but the GDL has an alternative syntax used to store and transfer the games: the Knowledge Interchange Format. It's based on a lisp and uses a prefix notation. Moreover :- operator is replaced with ⇐, & with and, negation ~ with not. The variables start with the ?. Below you can see an example of translation between Prolog:

and KIF:

4.1 Assignemnt 1

For every example below, tell if the KIF version is a faithful translation of the Prolog one.